#### Faculty Mentor

Jozsef Losonczy

#### Major/Area of Research

Mathematics

#### Description

An ideal I in a polynomial ring k[x1,...,xn] is a nonempty set closed under

addition satisfying hf _ I whenever h _ k[x1,...,xn] and f _ I. We say I is finitely

generated if there exist f1,...,fs _ I such that every element of I is of the form

h1f1+ΣΣΣ+hsfs for some h1,...,hs _ k[x1,...,xn]. By a fundamental result called

the Hilbert Basis Theorem, every polynomial ideal is finitely generated. Every

polynomial ideal has a special finite generating set called a Groebner basis.

A Groebner basis {g1,...,gt} is a set of polynomials in I such that the ideal

generated by the leading terms of polynomials in I coincides with the ideal

generated by the leading terms of g1,...,gt. Groebner bases were established

by Bruno Buchberger, and can be computed by the algorithm developed in

his 1965 PhD thesis. Groebner bases have many valuable applications in

commutative algebra. By their nature, they provide a means of ideal description.

They also give a definitive result in determining whether or not a

polynomial is a member of an ideal. Groebner bases are utilized in algebraic

geometry as well, such as in using elimination theory to solve a system of

polynomial equations, and in providing an implicit representation of a parametric

curve or surface. Lastly, Groebner bases can be used to describe the

results of certain operations on ideals, such as the intersection of two ideals

and the radical of an ideal, the latter being fundamental to the relationship

between ideals and varieties.

Applications of Groebner Bases in Commutative Algebra and Algebraic Geometry

An ideal I in a polynomial ring k[x1,...,xn] is a nonempty set closed under

addition satisfying hf _ I whenever h _ k[x1,...,xn] and f _ I. We say I is finitely

generated if there exist f1,...,fs _ I such that every element of I is of the form

h1f1+ΣΣΣ+hsfs for some h1,...,hs _ k[x1,...,xn]. By a fundamental result called

the Hilbert Basis Theorem, every polynomial ideal is finitely generated. Every

polynomial ideal has a special finite generating set called a Groebner basis.

A Groebner basis {g1,...,gt} is a set of polynomials in I such that the ideal

generated by the leading terms of polynomials in I coincides with the ideal

generated by the leading terms of g1,...,gt. Groebner bases were established

by Bruno Buchberger, and can be computed by the algorithm developed in

his 1965 PhD thesis. Groebner bases have many valuable applications in

commutative algebra. By their nature, they provide a means of ideal description.

They also give a definitive result in determining whether or not a

polynomial is a member of an ideal. Groebner bases are utilized in algebraic

geometry as well, such as in using elimination theory to solve a system of

polynomial equations, and in providing an implicit representation of a parametric

curve or surface. Lastly, Groebner bases can be used to describe the

results of certain operations on ideals, such as the intersection of two ideals

and the radical of an ideal, the latter being fundamental to the relationship

between ideals and varieties.